Can I patent a new product that utilises/recombines aspects of two other products that are both out of patent?
When a patent is examined whether another product that your invention relies on is patented or not is irrelevant. The test comes down to whether your invention is new and inventive given the things that have been known and used before your patent was filed.
If your combination is non-obvious, then it is likely patentable, provided it leads to some form of new and improved result (synergy – the whole is greater than the sum of its parts).
The question of whether there are other patents out there that your product relies on comes in when you enter the market. Having a patent does not give you the right to ignore other patents. You can hold a valid patent and infringe other patents. This is why doing freedom to operate (FTO) searching is important before entering the market with your product.